Can we make a Finsler metric complete by a trivial projective change?

Abstract

A trivial projective change of a Finsler metric $F$ is the Finsler metric $F+df$. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial projective change. The problem actually came from lorentz geometry and mathematical relativity: it was observed that it is possible to understand the light-line geodesics of a (normalized, standard) stationary 4-dimensional space-time as geodesics of a certain Finsler Randers metric on a 3-dimensional manifold. The trivial projective change of the Finsler metric corresponds to the choice of another 3-dimensional slice, and the existence of a trivial projective change that is forward and backward complete is equivalent to the global hyperbolicity of the space-time.

11 pages, one figure, submitted to the proceedings of VI International Meeting on Lorentzian Geometry (Granada)

References in corpus (5)

• On the energy functional on Finsler manifolds and applications to stationary spacetimes
• On the interplay between Lorentzian Causality and Finsler metrics of Randers type
• A note on the existence of standard splittings for conformally stationary spacetimes
• Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric
• Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness

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• An account on links between Finsler and Lorentz Geometries for Riemannian Geometers
• Metric structures associated to Finsler metrics
• On submaximal dimension of the group of almost isometries of Finsler metrics
• Width estimate and doubly warped product